The even-genus Rankin-product lifting conjecture

Let $m\geq 1$, and let $f$ and $g$ be Siegel modular forms of genus $2m$, of weights $k>2m$ and $l=k-2m$, with Satake parameters $(\alpha_0,\alpha_1,\ldots,\alpha_{2m})$ and $(\beta_0,\beta_1,\ldots,\beta_{2m})$. Even-genus lifting conjecture. There exists a Siegel modular form $F$ of genus $4m$ and weight $k$ with Satake parameters

$$\gamma_0=\alpha_0\beta_0,\quad \gamma_i=\alpha_i\ (1\leq i\leq 2m),\quad \gamma_{2m+j}=\beta_j\ (1\leq j\leq 2m).$$

The source further observes that the conjectural motives $M(Sp(f))\otimes M(Sp(g))$ and $M(Sp(F))$ have the same Hodge types and rank $2^{4m}$, which supplies motivation but not a proof of the lifting.